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Math Help - Series converge?

  1. #1
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    Series converge?

    Let a_n>0, \lim_{n\to\infty}a_n=a>0. Study the converegence or divergence of the series \sum_{n=1}^\infty (a/a_n)^n.

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    Re: Series converge?

    Quote Originally Posted by xinglongdada View Post
    Let a_n>0, \lim_{n\to\infty}a_n=a>0. Study the converegence or divergence of the series \sum_{n=1}^\infty (a/a_n)^n.

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    I think this series must diverge. If \displaystyle \begin{align*} a_n \to a \end{align*} then \displaystyle \begin{align*} \frac{a}{a_n} \to \frac{a}{a} = 1 \neq 0 \end{align*}.
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    Re: Series converge?

    But we may have the limit 1^\infty.
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    Re: Series converge?

    Quote Originally Posted by Prove It View Post
    I think this series must diverge. If \displaystyle \begin{align*} a_n \to a \end{align*} then \displaystyle \begin{align*} \frac{a}{a_n} \to \frac{a}{a} = 1 \neq 0 \end{align*}.
    For example,
    \lim_{n\to\infty}(1-1/\sqrt{n})^n=\lim_{n\to\infty}e^{-\sqrt{n}}=0.
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